Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 10.19880390272   b = 14.42222051019   c = 10.19880390272

Area: T = 52
Perimeter: p = 34.81882831562
Semiperimeter: s = 17.40991415781

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 10.19880390272
Height: hb = 7.21111025509
Height: hc = 10.19880390272

Median: ma = 11.4021754251
Median: mb = 7.21111025509
Median: mc = 11.4021754251

Inradius: r = 2.98769364763
Circumradius: R = 7.21111025509

Vertex coordinates: A[-3; 8] B[-1; -2] C[9; 0]
Centroid: CG[1.66766666667; 2]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 2.98769364763]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 135° = 0.78553981634 rad

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How did we calculate this triangle?

1. We compute side a from coordinates using the Pythagorean theorem

a = | beta gamma | = | beta - gamma | ; ; a**2 = ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (-1-9)**2 + (-2-0)**2 } ; ; a = sqrt{ 104 } = 10.2 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = | alpha gamma | = | alpha - gamma | ; ; b**2 = ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (-3-9)**2 + (8-0)**2 } ; ; b = sqrt{ 208 } = 14.42 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = | alpha beta | = | alpha - beta | ; ; c**2 = ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (-3-(-1))**2 + (8-(-2))**2 } ; ; c = sqrt{ 104 } = 10.2 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 10.2 ; ; b = 14.42 ; ; c = 10.2 ; ;

4. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 10.2+14.42+10.2 = 34.82 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 34.82 }{ 2 } = 17.41 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.41 * (17.41-10.2)(17.41-14.42)(17.41-10.2) } ; ; T = sqrt{ 2704 } = 52 ; ;

7. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 52 }{ 10.2 } = 10.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 52 }{ 14.42 } = 7.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 52 }{ 10.2 } = 10.2 ; ;

8. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 10.2**2-14.42**2-10.2**2 }{ 2 * 14.42 * 10.2 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14.42**2-10.2**2-10.2**2 }{ 2 * 10.2 * 10.2 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.2**2-10.2**2-14.42**2 }{ 2 * 14.42 * 10.2 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 52 }{ 17.41 } = 2.99 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.2 }{ 2 * sin 45° } = 7.21 ; ;




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